| L(s) = 1 | + (0.0735 + 0.417i)2-s + (1.58 − 0.706i)3-s + (1.71 − 0.622i)4-s + (0.766 + 0.642i)5-s + (0.411 + 0.607i)6-s + (0.939 + 0.342i)7-s + (0.809 + 1.40i)8-s + (2.00 − 2.23i)9-s + (−0.211 + 0.366i)10-s + (−0.254 + 0.213i)11-s + (2.26 − 2.19i)12-s + (0.357 − 2.02i)13-s + (−0.0735 + 0.417i)14-s + (1.66 + 0.475i)15-s + (2.26 − 1.89i)16-s + (−2.47 + 4.29i)17-s + ⋯ |
| L(s) = 1 | + (0.0520 + 0.294i)2-s + (0.913 − 0.407i)3-s + (0.855 − 0.311i)4-s + (0.342 + 0.287i)5-s + (0.167 + 0.248i)6-s + (0.355 + 0.129i)7-s + (0.286 + 0.495i)8-s + (0.667 − 0.744i)9-s + (−0.0669 + 0.115i)10-s + (−0.0768 + 0.0644i)11-s + (0.653 − 0.633i)12-s + (0.0990 − 0.561i)13-s + (−0.0196 + 0.111i)14-s + (0.430 + 0.122i)15-s + (0.566 − 0.474i)16-s + (−0.601 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.99083 - 0.116103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.99083 - 0.116103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.58 + 0.706i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| good | 2 | \( 1 + (-0.0735 - 0.417i)T + (-1.87 + 0.684i)T^{2} \) |
| 11 | \( 1 + (0.254 - 0.213i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.357 + 2.02i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.47 - 4.29i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.59 + 2.77i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.09 - 0.398i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.287 + 1.63i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (4.45 - 1.62i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (2.97 - 5.14i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.928 - 5.26i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.229 - 0.192i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.161 + 0.0586i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 1.59T + 53T^{2} \) |
| 59 | \( 1 + (5.71 + 4.79i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.35 - 0.492i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.146 + 0.830i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.692 + 1.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.74 + 13.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.26 + 7.19i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.28 - 12.9i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.43 - 4.21i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.16 + 6.00i)T + (16.8 - 95.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.11007396984048847048014321446, −9.047404438454943845209916090562, −8.218255463795487058711317569109, −7.53212442524064685523663267661, −6.63327770843709689856843735365, −6.03629873692012628803730600490, −4.82386974145329406970155006880, −3.43698482393396222274075888714, −2.40228304221069402233802155828, −1.56912288769147810949031358578,
1.70352427991732916174857126591, 2.48704547010643057662077670996, 3.63868998651653194675733804810, 4.48065253542197801328088254117, 5.66347135781978809420278844743, 6.92117540784119636393855005247, 7.52336858860605764991695553074, 8.511508483093926213050154641337, 9.183735038619602236430540459083, 10.10469241594346661171284415356
